Flatness of Dichroic Beamsplitters Affects Focus and Image Quality

Even though fluorescence microscopy has become a routine technique for many
applications, demanding requirements from technological advances continue to push the limits.
For example, today biological researchers are not only interested in looking at fixed-cell
samples labeled with multiple colors, but they are also interested in looking at the dynamics of
live specimens in multiple colors, simultaneously. Even certain fixed-cell imaging applications
are hampered by limited throughput of the conventional multicolor imaging techniques. Given
that the most popular multicolor fluorescence imaging configurations – including “Sedat” and
“Pinkel” [1] – are often too slow, new multicolor imaging approaches are evolving. The
schematic of such an imaging setup that enables “simultaneous” visualization of multicolor
images of a specimen is shown in Figure 1.

Figure 1: A simple setup for simultaneous visualization of two colors.

In this example, the dual-colored emission signal from the sample is split into two emission
light paths using a dichroic beamsplitter, and the signal in each of these channels is imaged
onto a CCD camera. An advantage of this configuration over conventional multicolored imaging
configurations is that the emission signal from multiple sources can be visualized
simultaneously. Due to the absence of any moving parts, such as a filter wheel, this approach
enables high-speed image acquisition, thus allowing for very fast cellular dynamics to be
imaged in multiple colors at the same time. The principal of splitting the emission signal into
different channels is already possible using commercially available products that allow for
multichannel imaging. For example, standard video adaptors available from microscope
companies can be attached to the emission port of a microscope. In another approach, multiple-
channel imaging systems like “Dual View” and “Quad View” (Photometrics) and “OptoSplit”
(Cairn Research) are based on the same principle, but instead of steering the emission signal of
different colors onto two different detectors, different regions of the same CCD chip
simultaneously image different colors.

These are just a few examples, which illustrate the application of an “imaging dichroic,” or a
dichroic in which not only the signal transmitted through the filter, but also the signal reflected off
of the dichroic is imaged onto a CCD camera. Considering the expected growth and popularity
of this imaging modality, optical components such as these dichroics merit a closer look. It may
be surprising to most users that a dichroic mirror (Fig.1) that transmits the emission signal of
one color and reflects the emission signal of another color is one of the most critical elements
that affect image quality. But in fact if this dichroic mirror is not sufficiently flat then significant
optical aberrations may be introduced and the imaging may be severely compromised.

Lack of sufficient flatness on a dichroic is not only a problem for imaging quality, but it can
also reduce the quality of the illumination light beam when the dichroic is used to reflect the
illumination light toward the sample. This problem is especially noticeable in Total Internal
Reflection Fluorescence (TIRF) microscopy [2], when a non-flat dichroic is placed in the
excitation light path.
2. Effects of dichroic flatness on imaging applications

An optical filter is generally comprised of multilayered thin-film coatings on the surfaces of a
plane, parallel glass substrate. In order to maximize transmission and reliability while minimizing
artifacts associated with multiple surfaces, single-substrate filters are designed with hard
coatings [1] on one or both sides (for example, Semrock’s ion-beam-sputtered filters). The glass
substrate is not always perfectly flat, especially after it is coated. The intrinsic stress of hard
glass coatings can be different from that of the substrate thus causing slight bending (or
curvature) of the substrate. Fortunately, this bending has no noticeable effect on light
transmitted through an optical filter at or near normal incidence.

For light incident at high angles of incidence, as is the case for a 45º dichroic beamsplitter,
the only appreciable effect of a bent substrate on transmitted light is a slight divergence of the
beam axis, similar to the effect of a small wedge in the substrate. But the quality of a beam
transmitted through a slightly bent substrate remains exceptional until the bending radius of
curvature is very small (much less than 1 meter). Therefore, in such a situation the focused spot
size of a laser beam or the aberrations associated with an imaging beam transmitted through a
bent filter are similar to their values with no filter present. This is the typical imaging
configuration in epifluorescence microscopy where the emission signal is transmitted through
the dichroic.

However, a bent filter substrate can have a significant impact on the quality of reflected light.
This situation is depicted in Figure 2. As shown in diagram, due to the curvature of the dichroic
two main effects occur: (1) the position of the focal plane shifts and (2) the size (or shape) of the
focused spot changes. Either (or the combination) of these two effects can significantly
compromise the image quality.

Figure 2: A bent dichroic introduces focal shift and a change in the spot size.

Generally a small shift of the focal plane is not a problem, because a lens or a camera
adjustment can be made to compensate for the shift. For example, when an imaging beam is
split by a dichroic onto two different cameras (as in Fig. 2), a minor adjustment to the position of
the camera associated with the reflected light path is generally possible. However, such
mechanical adjustments can not compensate for a change in the size or shape of the focused
spot when the flatness of the dichroic is significantly compromised. Therefore, the quality of an
image formed using reflected light off of a non-flat dichroic can be significantly worse than the
ideal, diffraction-limited image formed using a perfectly flat dichroic.

In our experience measuring actual dichroic beamsplitters and using optical ray-tracing
analyses, the dominant contribution to non-flatness of hard-coated thin-film filters is spherical
curvature, and the dominant optical aberration that results when light is reflected off such a filter
at 45º is astigmatism. A collimated beam of light focused by an astigmatic lens or mirror exhibits
two distinct focal planes – called the tangential and sagittal planes – and the focused spot in
each plane is asymmetric, as if it is focused along only one axis of the plane, but is out of focus
along the other axis. Between these two planes is a third plane that contains the smallest
circular focused spot, which is commonly called the “circle of least confusion.” Though
symmetric, this spot is larger than the spot that would be obtained in the absence of
astigmatism in the focusing lens or mirror, thus resulting in blurring of the image.

To demonstrate and quantify this effect, measurements were made using a number of
different reflecting dichroics and mirrors with the experimental setup shown in Figure 3. Four
filters were chosen with a wide variation of non-flatness arising from differing radii of curvature –
the residual flatness (after removing the best-fit spherical error) for all filters were excellent.

Figure 3: An experimental setup to evaluate the effect of flatness of a reflecting surface on imaging. A
laser beam (532 nm) was spatially filtered using a 10μm pinhole before collimation with a single lens.
The beam diameter of the collimated beam was adjusted using a variable aperture. Reflecting surfaces of
different radii of curvature (3414 m, 158 m, 84 m, and 7 m) were used to steer the beams of different
diameters (11 mm or 2.5 mm) using a 300 mm focal length plano-convex tube lens onto a CCD camera.
The CCD camera was mounted on a linear translation stage for imaging the focal spots.

Images of the focused spot obtained from each of the four filters are shown for an 11 mm
diameter beam in Figure 4. For each filter, the images were acquired at the position of best
focus, as well as at the tangential and sagittal focal planes. The in-focus image quality obtained
using reflection from a very flat surface (Fig. 4, top row) can deteriorate significantly (Fig. 4,
bottom row) due to the astigmatism introduced by a non-flat reflecting surface. The degree of
astigmatism decreases (Fig. 4, third row) or almost vanishes (Fig.4, second row) when a
relatively flatter reflecting surface is used. These images clearly demonstrate that flatness of the
dichroic has a significant effect on the reflected image quality. It should be noted that in these
results not only the size and shape of the focal spot change, but the focal plane location is also
different for the different radii of curvature of the reflecting surfaces (see Section 3 below).

Figure 4: Images of the focused spot from an 11 mm diameter beam reflected off of dichroics of different
flatness and focused onto a CCD detector using a 300 mm focal length tube lens. Images were acquired
for the best-focus position as well as at the tangential and sagittal focal planes. Size bar for 7 m radius of
curvature dichroic is 1 mm; all other size bars are 100μm. The intensity distributions from these images
are plotted as 3D surface plots in Figure 10 (see Appendix).

It should be noted that the diameter of the illumination beam that impinges on a dichroic is
generally much smaller than an imaging beam (emission signal) in most applications. For
example, a laser beam reflecting off of a dichroic beamsplitter in a TIRF microscope is generally
only about 1 mm to perhaps at most several mm in diameters. Since the focal spot size is
significantly determined by the beam diameter (see Section 3), further experiments were
conducted to observe the effect on spot size when reflecting surfaces of different radii of
curvatures were used. Results based on a 2.5 mm diameter beam are shown in Figure 5. As
seen in this figure, the size (and shape) of the best-focus spot does not appear to change nearly
as dramatically as it does with the larger diameter beam in Fig. 4. The variation in spot sizes is
also practically negligible between tangential and sagittal focal planes.

Figure 5: Images of the focused spot from a 2.5 mm diameter beam reflected off of dichroics of different
flatness and focused onto a CCD detector using a 300 mm focal length tube lens. Images were acquired
for best-focus position as well as at the tangential and sagittal focal planes. Size bar is 100μm. The
intensity distributions from these images are plotted as 3D surface plots in Figure 11 (see Appendix).

It was pointed out above that in addition to degradation of the spot size and shape, reflection
off of a non-flat dichroic also causes an appreciable shift in the focal plane relative to that
obtained from reflection off of a perfectly flat mirror. Even though a small focal shift can be
compensated in many situations (as noted above), there are cases in which a focal shift cannot
be tolerated. For example, when a laser beam is reflected off of a dichroic beamsplitter in a
TIRF microscope, the beam must be focused at the back focal plane of the objective. A
significant focal shift or a change in the focal spot size caused by a bent dichroic placed in the
excitation light path can make it difficult to achieve TIRF, especially if the microscope has a
limited ability to adjust the collimation of the laser beam. A non-flat dichroic can also cause a
similar problem in a structured-illumination microscope with broadband excitation, in which the
mask must be imaged in the excitation light path onto the sample. Due to mechanical
constraints it might be impossible to image the grid onto the sample plane, or at least to achieve
a good image with the objective positioned to acquire a good image of the emitted light. A
similar problem can also occur in confocal instruments that require sufficiently flat dichroics to
enable imaging of pinholes at desired focal planes.
3. Analysis of the effects of non-flat dichroics

Given that a dichroic mirror may be used to steer the excitation light or the emission signal
to a desired location, evaluation measures are needed to find out whether a dichroic mirror is
sufficiently flat for a given application. For example, as observed in Section 2, beams of different
diameters may require different flatness specifications. It should be noted that since thin-film
coatings can themselves cause bending of the dichroic mirror, the substrate flatness before
coating is not a sufficient criterion for evaluation of the performance of the filter. Therefore the
flatness requirements for a dichroic should be specified for the finished, coated substrate.Focal plane shift

When light is reflected off of a spherically curved surface with radius R, the light is focused
as if it were transmitted through a lens of focal length R/2. If the light is subsequently focused by
a different lens of focal length f (and where we assume f << R/2), the reflection off of the
curved surface causes the focus to shift by an amountrelative to the focus position resulting
from reflection off of a perfectly flat surface. Based on a simple first-order optics approximation,
the relative focal shift is. Thus, to maintain a relative focal shiftbelow a
certain required value, the radius of curvature of the bent filter surface should exceed

For example, suppose a laser beam is reflected off of a dichroic beamsplitter before being
focused to a spot at the sample plane by a 40X, 0.75 Numerical Aperture (NA) microscope
objective, which has a focal length of f= 5 mm (assuming a 200 mm focal length of tube lens),
and suppose we want to keep the focal shiftbelow 5μm, or the relative focal shift below
0.1%. Then the radius of curvature R of the dichroic must be greater than about 10 meters.

When working with Gaussian beams (often an excellent model for a laser beam), the depth
of focus is quantified by the “Rayleigh Range,” which is the distance from the beam waist
(smallest focused spot) to the point where the waist has increased by a factor of.
Practically, the Rayleigh Range is a good measure of depth of focus because a beam (or image
point) focused to a location within one Rayleigh Range of the waist still appears to be in focus. If
we use the criterion that the focal shift should be less than one Rayleigh Range in order to be
considered negligible, then the radius of curvature of a reflecting filter in front of a focusing lens
should exceed

where D is the diameter of the beam at the focusing lens andis the wavelength of light. Note
that the required radius of curvature is independent of the focal length of the lens. As an
example, for a 1 mm diameter 488 nm laser beam, the radius of curvature of the dichroic should
be greater than about 3 meters. Semrock’s laser-quality dichroic beamsplitters are generally
specified with a maximum beam diameter, D, for which the focal shift is less than one Rayleigh
Range at the edge wavelength of the beamsplitter. For instance, the Di01-R488-25×36 is a
BrightLine® Laser Dichroic, and it is specified to have less than one Rayleigh Range of focal
shift for a 488 nm laser beam as large as 2.5 mm in diameter. For a 1 mm diameter beam, the
focal shift would be less than 1/6th of the Rayleigh Range.

As an imaging example, consider a 40X, 0.75 NA microscope objective for which the beam
diameter is about 7.5 mm. Thus the radius of the dichroic must exceed about 160 meters in
order to observe no noticeable focal shift. Most dichroics are not this flat, and thus in general
there is a noticeable focal shift for light reflected off of a dichroic relative to a perfectly flat mirror.
However, as pointed out above, this type of focal shift is generally easily accommodated.

Many of the aberrations in imaging that result from wedge error and wavefront errors in thinfilm
interference filters [3] can be tightly controlled by selecting the best available glass
substrates. In fact, the simplified single-substrate approach enabled by ion-beam sputtering
technology allows for the utmost in control of substrate-induced aberrations. However the
inherent bending introduced during the thin-film coating manufacturing process can alter the
original flatness specifications of the substrate. Fortunately, this bending can also be controlled
with appropriate choice of coating materials, substrate, and overall filter design. Furthermore,
precisely controlled manufacturing processes that produce highly repeatable coatings are
critical to achieving flat filters. The resulting dichroic beamsplitters are ultimately limited by a
very small amount of spherical bending due to manufacturing uncertainties.

Considering the astigmatism introduced by a non-flat dichroic with a small amount of
spherical bending, the location of the two main focal planes (Figs. 4 & 5) are given by the
tangential and sagittal focal lengths, fT and fS, respectively [4]:

where fTL is the focal length of the tube lens (see schematic in Fig. 3). An obvious way to
quantify the astigmatism is to measure the separation between the tangential and sagittal focal
planes – the larger the separation, the greater the astigmatism. Using Eqs. (3) and (4), we can
generate the curves shown on the graph in Figure 6, illustrating the dependence of this
separation on radius of curvature for different tube lens focal lengths.

Figure 6: A change in the radius of curvature of the dichroic mirror affects the degree of astigmatism.fT and fSare the tangential and sagittal focal lengths, respectively, and fTLis the focal length of
the tube lens.

Specifically, for a 300 mm focal length tube length (corresponding to the experimental
imaging conditions used to produce the results in Figs. 4 and 5) the following table summarizes
the locations of the tangential and sagittal focal planes for different radii of curvature of the
reflecting surfaces.

Table 1: A change in the radius of curvature of the dichroic mirror can significantly change the
location of the focal spot.

Spot size change

As noted above, the non-flatness of the dichroic mirror not only changes the location of the
focal plane, but also the size of the image in the focal plane may be significantly affected. For an
astigmatic beam, a simple first-order optics calculation can be used to show that the geometric
spot size, or diameter of the “circle of least confusion,” DC, is given by

where D is the diameter of the beam and R is the radius of curvature of the reflecting surface
(Fig. 3). Eq. (5) indicates that the beam diameter is directly proportional to the spot size. For the
experiments presented in Figs. 4 and 5, the calculated geometric spot sizes are given in Table 2.

Table 2: Radius of curvature of the reflecting surface and the beam diameter (in
Fig. 3) significantly affect the spot size for a given imaging setup. This table gives
the theoretical geometric spot size limited by a simple ray optics analysis, and
does not take into account the diffraction limit.

Even though this simple geometric calculation can provide a reasonable estimate of spot
size for relatively larger sizes, smaller spot sizes are limited by diffraction of light due to its wave
nature. The size of the Airy disc [5] is a widely accepted criterion for the evaluation of the
diffraction-limited spot size. For the experimental conditions applicable to Figs. 3 – 5, the ideal
diffraction-limited spot sizes are 77.88μm and 17.7μm for the 2.5 mm and 11 mm diameter
beams, respectively. The variation of geometric spot size as a function of radius of curvature for
various beams diameters is plotted in Figure 7. As shown in this figure, diffraction-limited spot
size limits the requirement of the sufficient radius of curvature for an application.

Figure 7: Theoretical geometric spot size changes as a function of the radius of curvature. The
horizontal dotted lines denote the size of the Airy disc (diffraction limit) and the vertical dotted lines
represent the corresponding desired minimum radius of curvature of the reflecting surface in Figure 3 to
ensure that the astigmatism-limited spot size is below the diffraction-limited spot size. The focal length of
the tube lens is assumed to be 300 mm.

In order to ensure that the spot size is not appreciably affected by astigmatism, the diameter
of the spot size determined strictly by geometric optics at the “circle of least confusion” should
be comparable to or smaller than the diffraction-limited spot size (Airy disc). This criterion leads
to the following requirement for the radius of curvature of the dichroic,

where D is the beam diameter andis the wavelength of light. Once the beam diameter and
the radius of curvature are known, the flatness specification can be specified in terms of waves
(for a given beam diameter). The center to edge deflectionis given by (to second order)

Therefore, in order to ensure that there is no noticeable effect of astigmatism, the flatness of the
dichroic specified in terms of center to edge deflection over the beam area is approximately
given by

where is the wavelength of light that is reflected off of the dichroic in the system (and where
we have approximated ).

The above analysis considers approximately only the effects of astigmatism, which as we
have pointed out is the dominant aberration. We can model more thoroughly what results when
light is incident at 45º and is subsequently reflected off of a dichroic beamsplitter with a slight
bend. Third- and higher-order aberrations (including the dominant astigmatism) degrade the
quality of a focused spot size (for a collimated beam) after a focusing lens or of an image after
an imaging lens. Using standard optical modeling software (like ZEMAX or Code V® by Optical
Research Associates), we can calculate the geometric RMS spot size in the absence of
diffraction that results from reflection off mirrors with varying radii of curvature.

The plot in Figure 8 is based on a typical epifluorescence microscope configuration,
assuming a perfect point source at the sample location, imaged onto the image plane (e.g.,
CCD surface) by an ideal 40X, 0.75 NA objective and a tube lens with a 200 mm typical focal
length. The resulting beam diameter is 7.5 mm. The reflection off of the dichroic is assumed to
occur mid-way between the objective and the tube lens. The field of view of the system is
assumed to be limited by a 20 mm diameter field size at the camera plane. The light is
assumed to have a wavelength of 510 nm (peak of GFP emission). For comparison, the
diffraction-limited spot size that would result from a perfect objective and tube lense and a
perfectly flat dichroic is 16.6 μm.

Figure 8: Spot size changes with a change in the radius of curvature of the dichroic. This example
shows a ray optics calculation of the RMS spot size (absent diffraction) in a typical epifluorescence
microscope configuration, with a 40X, 0.75 NA objective and a 200 mm focal length tube lens.

For the above example, the effects of ray aberrations (dominated by astigmatism) that result
from the non-flat dichroic become less than that of diffraction when the radius of curvature
exceeds about 50 meters. Note that the simple astigmatism analysis summarized by Eq. (6)
suggests the radius of curvature for this case should be larger than about 64 meters, showing
close agreement between the detailed ray aberration analysis and the simple astigmatism
model.

The required minimum radius of curvature from Eq. (6) for a number of other objective-tube
lens combinations that are common in fluorescence microscopes are summarized in Figure 9.
The required minimum radii vary from a few tens of meters for the higher magnification
objectives (with smaller beam diameter) to as high as about 50 to 100 meters for the lower
magnification objectives (with larger beam diameter). Note that to calculate the beam diameter
for an objective with a specified magnification and NA, it is necessary to know what tube lens
focal length is assumed to achieve the stated magnification. The beam diameter is given simply
by [6]

where M and NA are the objective magnification and numerical aperture, and fTL is the tube lens
focal length.

Figure 9: Desired radii of curvature of an “imaging dichroic” for different objective and tube lens
combinations (Magnification x NA of each objective is listed). Tube lengths of (left to right) 160, 200, 200,
200, 160, 200, 160, 200, and 160 mm were assumed. Calculations based on Eq. (6) where the beam
diameter D is calculated using Eq. (9). Results correspond to an emission signal at a wavelength of 510
nm (GFP emission).
4. Comparison of actual images

Images of fixed cells were acquired using a modified microscope setup that was designed to
evaluate the effect of dichroic flatness on image quality. The emission optics (including the
CCD camera) of a standard epifluorescence microscope was removed from its body and the
reflecting surfaces of different radii of curvature (used for the experiments reported above in
Figs. 4 and 5) were used to steer the emission signal from the microscope onto the CCD
detector. The principle of operation of this imaging setup was similar to the one described in
Figure 3, except that the collimated beam was generated by the objective lens used to collect
the fluorescent signal from the sample. For each reflecting filter, the objective focus was
adjusted to achieve the clearest possible image. The resulting images are shown in Figure 10.
When the reflecting surface does not have a sufficiently large radius of curvature (right image),
significant blurring is introduced in the acquired image due to astigmatism. However when a
dichroic of sufficient flatness is used (as demonstrated in the left three figures) then imaging
quality is not compromised.

Figure 10: Images of F-actin in bovine pulmonary artery endothelial cells (Fluo Cells prepared slide #1
from Invitrogen) captured using a modified Olympus BX41 microscope. The emission signal from the
microscope was reflected using surfaces of different radii of curvature. A 40X, 0.75 NA objective and 200
mm focal length tube lens were used in this experiment. The desired radius of curvature of the dichroic in
order to avoid image degradation is about 64 meters. Size bar is 2 μm.
5. Concluding remarks

Reflected image quality can be worse than the ideal diffraction-limited response for dichroics
that are not perfectly flat, though it should be noted that the true spot size at the image plane
can be appreciably larger than the diffraction-limited spot size in an actual system.
Nevertheless, care should be taken to select properly optimized, flatter dichroic beamsplitters
when working with reflected light. Dichroics designed to reflect laser light (“laser dichroics”) are
generally flat enough to ensure negligible focal shift for laser beams up to several mm in
diameter. Dichroics designed to reflect imaging beams (“imaging dichroics”) have the most
extreme flatness requirements, since they must effectively eliminate the effects of astigmatism
for beams as large as 1 cm or more.
6. References